# How much did the universe expand since recombination time? [closed]

Assuming adiabatic expansion, and a temperature at recombination time of $$2700^{\circ} C$$, how much has the universe expanded if the temperature today is $$2.7K$$?

My idea is to use

$$\frac{dQ}{dt}=A\sigma T^4$$

where $$\frac{dQ}{dt}=0$$, A is the blackbody surface area, and T is the temperature. The way I did it was since the expansion is adiabatic (subscript $$0$$ for recombination time and $$f$$ for today):

$$A_0\sigma T_0^4=A_f\sigma T_f^4$$

and solving for $$\frac{A_f}{A_0}$$:

$$\frac{A_f}{A_0}=\frac{T_0^4}{T_f^4}=\frac{(2973K)^4}{(2.7K)^4}=1.47\times10^{12}$$.

So the surface area of the universe is $$1.47\times 10^{12}$$ larger now than at recombination time. Is this correct?

## closed as off-topic by JMac, G. Smith, GiorgioP, John Rennie, Kyle KanosApr 2 at 11:39

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• It is more useful to think about the energy density. The energy density of a gas of photon also goes as $\propto T^4$. How does the energy density of relativistic matter depend on the scale factor? This will tell you how $T$ depends on the scale factor. – octonion Apr 1 at 18:24

The temperature $$T$$ of the cosmic background radiation is given by the energy of the photons, which redshift proportionally to the expansion of the Universe. That is, $$a T = a_0 T_0,$$ where $$a$$ is the scale factor, and subscript $$0$$ denote values today. Hence, when $$T$$ drops to a fraction $$T_0 / T = 2.7\,\mathrm{K} \,/\, 3000\,\mathrm{K} \simeq 1/1100$$, that means that the Universe has expanded by a factor $$a_0/a = 1100$$.

Usually we define the scale factor today to be unity, and just say $$a$$ was $$1/1100 \simeq 9\times10^{-4}$$ at that time. The redshift $$z$$ of the photons received from that time is then $$z+1=1/a\simeq1100$$.

This expansion is in all directions. So distances between galaxies (that are not so close that they are gravitationally bound) increase by a factor $$a^{-1}$$, the surface area of the observable Universe increases by a factor $$a^{-2} \simeq 1.2\times10^6$$, and the volume of the Universe increases by a factor $$a^{-3} \simeq 1.3\times10^9$$.

Note that when you talk about how much the Universe has expanded, it is common to quote the "linear" expansion, i.e. the scale factor $$a$$, although in principle the volume factor $$a^3$$ might strictly be a more appropriate term. In contrast, I've never seen anybody talk about how much the surface area of the observable Universe has expanded; $$a^2$$ doesn't really bear any physical significance, but $$a$$ and $$a^3$$ does, as they describe the change in distances between objects, and volumes and — in particular — densities of objects, respectively.

• “Usually we define the scale factor today to be unity, and just say 𝑎=1100.” The universe had a smaller scale factor in the past, not a larger one. You got your ratio backward. – G. Smith Apr 1 at 19:32
• It is the redshift, not the scale factor, that is estimated to be about 1100. – D. Halsey Apr 1 at 21:09
• Okay I'm sorry, I was a bit too fast here. Thanks @G.Smith – pela Apr 1 at 21:25
• …and @D.Halsey :) – pela Apr 1 at 21:25
• Downvoted? But why? – pela Apr 2 at 11:54